By Leon O. Chua
Quantity III maintains the author's quest for constructing a pedagogical, self-contained, but rigorous analytical idea of 1-D mobile automata through a nonlinear dynamics viewpoint. utilizing rigorously conceived and illuminating colour pics, the worldwide dynamical behaviors of the 50 (out of 256) neighborhood principles that experience now not but been coated in Volumes I and II are uncovered through their stunningly revealing basin tree diagrams. The Bernoulli -shift dynamics stumbled on in quantity II is generalized to carry for all 50 (or 18 globally an identical) neighborhood principles through advanced and hyper Bernoulli wave dynamics. particular international kingdom transition formulation derived for principles 60, ninety, one hundred and five, and one hundred fifty show a brand new scale-free phenomenon. the main staggering new outcome unveiled during this quantity is the Isle of Eden came across hidden in so much (almost 90%) of the 256 neighborhood principles. Readers are challenged to seek for long-period, remoted Isles of Eden. those are infrequent gem stones ready to be chanced on.
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Additional info for A Nonlinear Dynamics Perspective of WolframÂ’s New Kind of Science: (Volume III) (World Scientific Series on Nonlinear Science, Series a) (World Scientific ... Science, Series a Monographs and Treatises)
Sample text
5 φn - 1 1 (Continued ) 10:6 22 , L = 5 May 6, 2009 Table 15. 96875 (a) Period-1 Attractor : ρ 1 = 64 −− 60 39 Transient phase 15 (Continued ) 10:6 22 , L = 6 May 6, 2009 Table 15. 890625 42 20 19 41 99 0 1 2 3 4 5 6 7 8 15 ch01 43 10:6 (a) Period-1 Attractor : 84 22 , L = 7 15 May 6, 2009 Table 15. 109375 Period-7 Isles of Eden : ρ 3 = 2 128 −− 10:6 22 , L = 7 May 6, 2009 Table 15. 6015625 197 168 253 0 8 107 140 104 34 146 68 127 193 164 128 16 25 182 190 215 22 208 162 116 169 214 134 23 43 46 49 200 232 97 167 137 176 13 19 203 61 77 69 145 188 73 94 113 109 122 56 152 125 1 131 229 11 89 191 238 28 211 42 71 254 119 82 235 Transient phase 37 223 84 226 142 212 154 149 178 Gallery 22 - 7 0 1 2 3 4 5 6 7 8 9 10 149 ch01 29 163 184 139 (a) Period-1 Attractor : 10:6 166 202 (Continued ) 22 , L = 8 172 May 6, 2009 Table 15.
Observe that local rules 6 , 9 , 11 , 14 , 27 , 35 , 38 , 43 , 56 , 57 , 58 , 134 , 142 , and 184 have two robust Bernoulli attractors, whereas local rules 25 and 74 have three robust Bernoulli attractors. Observe from Table 10 that only five rules listed in Table 10 11 , 14 , 15 , 43 and 142 have a negative sign for β. The space-time evolution patterns of these five rules are generated by following the same procedures as the other rules (shift left by σ bits if σ > 0, or shift right by |σ| bits if σ < 0, every τ iterations), and then complementing (change color of all bits) the resulting bit string.
The time-2 map φn−2 → φn consists of β = 2σ1 = 32 parallel red Bernoulli lines with slope 2σ1 = 32, or equivalently, to β = 2|−σ2 | = 32 parallel blue Bernoulli lines with slope 2σ2 = 1/32. Observe that the two red dots now fall on the diagonal of the time-2 map, as expected of period-2 orbits. Again, β > 0 because the slope of each red (or blue) Bernoulli line is positive. For ease of visualization, we have displayed the space-time pattern using bit strings with double the length, namely, 2L = 10, which corresponds to shifting around the period-2 ring twice.